Every finite point set P in the real plane carries a collection of distinct distances at each of its pins x in P. Two questions attach to this data. How few distinct distances can a single pin see, and how many pins can be bad at once. The second question is the sharper one, and the answer below is that bad pins can never occupy a positive fraction of a large configuration. For x in P, write D_x(P) = { |y - x| : y in P, y != x }. Erdos asked in 1957 whether some pin always sees at least n^(1 - eps) distinct distances, for every fixed eps > 0. The question is Problem 16 of Some unsolved problems (Michigan Math J 4(3):291-300, 1957). ./erdos1957-unsolved-prob.pdf "Let S be a finite point set in the plane, let N(S) denote the number of different values which occur among the distances, and f(n) the least value that N(S) can have. It can be conjectured that f(n) > n^{1-eps} for large n and, further, that there exists a point p_i at which more than n^{1-eps} different values occur among the distances." Some unsolved problems, Problem 16 (1957) The theorem below proves a strictly stronger statement: at all but o(n) pins the count reaches n^(1 - eps). The right object to track is not just the count but the multiplicity of any single distance at a pin. For distinct x, y in P define k_P(x,y) = #{ z in P \ {x} : |z - x| = |y - x| }. So k_P(x,y) counts every point at the same distance from x as y is, including y itself. Different pairs use their own source point, and two pairs with the same numerical distance but different sources are treated separately. For n >= 2 and s > 0 put F_n(s) = sup_{ P subset R^2, |P| = n } ( #{ (x,y) in P^2 : x != y, k_P(x,y) >= n^s } / ( n (n - 1) ) ). Theorem. For every fixed s > 0, F_n(s) -> 0 as n -> infinity. The convergence is uniform: no separation, general position, or coordinate hypothesis is imposed, and the constants only depend on s. The exponent s is fixed; the theorem supplies no explicit rate of decay. Corollary. For every fixed eps > 0, sup_{ P subset R^2, |P| = n } ( #{ x in P : |D_x(P)| < n^(1 - eps) } / n ) -> 0. In particular every sufficiently large n-point planar set has a pin determining at least n^(1 - eps) distinct nonzero distances. Proof of the corollary. For eps >= 1 the assertion is trivial; every pin with n >= 2 sees at least one distance. Suppose 0 < eps < 1 and take s = eps/2. A pin with fewer than n^(1 - eps) distance classes can place at most n^(1 - eps + s) neighbours in classes of size below n^s, so it contributes at least n - 1 - n^(1 - eps/2) pairs counted by F_n(s). If there are b such bad pins, then for all large n b/n <= (n-1) F_n(eps/2) / ( n - 1 - n^(1 - eps/2) ) -> 0. The bound is uniform in P. The corollary is nearly sharp. If P consists of the vertices of a regular polygon together with its centre, the centre sees a single nonzero distance. Exceptional pins are unavoidable; the theorem only says that such pins cannot fill a positive fraction of the population. History and external context. Erdos opened both the distinct-distance and the repeated-distance questions in 1946. ./erdos1946-amm-distances-n.pdf The original note introduces the quantity that later became the object of study. Denote by f(n) the least number such that any set of n points of the plane contains some point that determines at least f(n) distinct distances to the remaining points. The note points out that the integer square lattice provides sets for which the union of all such distances has size O(n / sqrt(log n)). The present question fixes one source and measures the count at that source. In that reformulation the lattice is not the extremal example; a symmetric configuration close to a regular polygon can make a single pin see just one distance. Guth and Katz proved that every planar n-point set determines at least a constant times n / log n distinct distances. ./guth-katz-2015-annals.pdf "In this paper, it is shown that a set of N points in R^2 has at least c N/log N distinct distances, thus obtaining the sharp exponent in a problem of Erdos." On the Erdos distinct distance problem in the plane, Ann. of Math. 181 (2015) Their conclusion concerns the union over all sources; the pinned version fixes the source and asks for the worst single-source count in the same configuration. Pinned bounds grew historically by combining incidence geometry with progressively sharper counting. Solymosi and Toth obtained the order n^(6/7). ./solymosi-toth-2001.pdf "It is shown that every set of n points in the plane has an element from which there are at least c n^{6/7} other elements at distinct distances, where c > 0 is a constant." Distinct Distances in the Plane, Discrete Comput. Geom. 25 (2001) Katz and Tardos pushed every exponent below (48 - 14 e) / (55 - 16 e) = 0.864137... in their paper A new entropy inequality for the Erdos distance problem. ./katz-tardos-2004.pdf Each of these papers guarantees that some pin in every configuration is good. The sharper pinned conjecture asks for order n / sqrt(log n) distances at a pin; the introduction to Bisectors and pinned distances documents the distinction between the real planar problem and pinned-distance theorems over other fields. ./lund-petridis-2020.pdf "Erdos also made a refinement of this conjecture. This refinement is that among any set A of N points in R^2, there must occur a point a in A such that the remaining points of A are at Omega(N log^{-1/2}(N)) different distances from a. This is often called the pinned distances conjecture." Bisectors and pinned distances, Discrete Comput. Geom. 64 (2020) The almost-every-pin corollary above resolves the weak pinned conjecture in the form in which it appears as Conjecture 4.7 of Generalised Erdos distance theory on graphs. ./dewar-etal-2025-weak-pinned.pdf Outline of the argument. Assume the theorem fails. Then a positive density of ordered distinct pairs have sources whose distance fiber is as large as n^s. Record these pairs as a directed graph, and let theta be the limiting edge density. A deletion procedure on the graph forces both a density bound from below and a fiber size bound from above, giving good control of large and small subsets. Work inside a number field K subset C that contains the coordinates of every point. Write x = (u,v) and set Z_1(x) = u + iv and Z_2(x) = u - iv, with i = sqrt(-1). The squared distance factors exactly: |x - y|^2 = ( Z_1(y) - Z_1(x) ) ( Z_2(y) - Z_2(x) ). The product formula in K, applied to both coordinates, balances the logarithms of these differences across all places, so on a single fiber the sum of the two coordinate depths is a fixed constant. To turn logarithms of distances into integers, nested partitions are built from randomly shifted square grids at each complex place. The grid depth differs from the raw logarithmic depth by a small error Delta whose distribution is independent of the pair and which has an exponential tail. Replacing any cell with more than half the points by its complement keeps the cell size balanced and produces an exact additive identity: the total overlap of two distinct points, summed over all levels, all coordinates and all places, equals a function of the first point plus a function of the second point plus a constant. That identity produces a nonnegative scale W, the expected overlap of two distinct uniform points plus a correction for giant complements. Two facts pin down W. The variance estimate compares the fiber law and the uniform law cell by cell and shows that cells of size at most n^(1-c) carry only o(W+1) of the overlap; the same variance estimate compares the incoming-neighbour law at each target with the uniform law. Then W either stays bounded or grows without bound along some subsequence. In the unbounded case, cells of mass bounded away from zero define weighted trees; transporting mass along these trees uses a classical distance formula for trees, and comparison of the fiber law with the uniform law turns the fiber equation into an impossibility for three independent uniform vertices. In the bounded case, one complex embedding is normalized so that the two coordinate laws have nonatomic weak limits; the variance estimate preserves the fiber equation in the limit and yields a family of Mobius maps between the two limits. A short order-statistics argument rules out any such family when both limit laws are nonatomic. Both alternatives are impossible, hence the theorem. The proof of the main theorem begins. Assume for the rest of the argument that the conclusion fails at some exponent: for at least one positive t the numbers F_n(t) do not tend to zero. Define Theta(t) = limsup_{n -> infinity} F_n(t), s_* = sup { t > 0 : Theta(t) > 0 }. Each F_n, and hence Theta, is nonincreasing. And F_n(t) = 0 vanishes for t >= 1 because a distance class at a pin has at most n - 1 points. So 0 < s_* <= 1. Assuming the theorem fails means Theta(t) > 0 for some positive t, hence s_* > 0. Fix 0 < c < s_*/3, max { c, (1-c) s_* } < s < s_*, c < gamma < s, choosing s to be a continuity point of Theta. Such an s exists because a monotone function has at most countably many discontinuities. Set theta = Theta(s) > 0; positivity follows from s < s_* and monotonicity of Theta. All asymptotic estimates below are with respect to the sequence of configurations constructed in the next step; their constants may depend on (c, s, gamma, theta) but not on n or on the configurations. Two types of probability laws at a graph. Once the directed graph E of large-fiber pairs is chosen, several probability measures attached to it will appear repeatedly. The piece of terminology that makes the pair laws usable is this: two edges lie in the same fiber when both share their source and their squared distance. Write x_e for the source of a fiber e, F_e subset of P minus {x_e} for its target set, and k_e = |F_e|. Let d^-(y) denote the number of incoming edges at y. The laws are these. alpha: uniform probability measure on P. pi: uniform on the n(n-1) distinct ordered pairs in P. q: uniform on E, with source and target marginals q_X and q_Y. mu_e: uniform on F_e. rho_y: uniform on the incoming neighbours of y, defined when d^-(y) > 0. To sample q, choose a fiber e with probability k_e / |E| and then choose y with law mu_e, setting x = x_e. Equivalently, sample y with law q_Y and then x with law rho_y. These laws are attached to the finite graph; the laws do not depend on any of the absolute values introduced later. The extraction of the graph from the failure of the theorem controls not only its density and fiber sizes but also how much of a fiber can sit inside a small subset of vertices. That last control is what lets fibers be compared with the uniform distribution. Tail lemma. If r_N -> s and M_N -> infinity, then sup_{ m >= M_N } ( F_m(r_N) - theta )_+ -> 0, where the supremum is over integers m >= 2 and (u)_+ = max { u, 0 }. Proof. Fix eps > 0. Continuity of Theta at s gives delta > 0 with Theta(s - delta) < theta + eps/2. By the definition of limsup, F_m(s - delta) <= Theta(s - delta) + eps/2 for all large m. For all large N, r_N >= s - delta, and every m >= M_N then lands in this range. Monotonicity of F_m in its argument gives F_m(r_N) <= theta + eps. Extremal graph proposition. Under the present supposition there is a sequence of finite sets P subset of R^2 of real algebraic points, with n = |P| -> infinity, and nonempty directed graphs E on P, with these properties. Their densities d_n = |E| / [n (n-1)] tend to theta, and every fiber has k_e >= n^(s - o(1)), uniformly in e. There is a fixed constant B_0 such that, pointwise, q <= B_0 pi, q_X, q_Y <= B_0 alpha, (q_X + q_Y)/2 >= (1 - o(1)) alpha, and q_Y { y : d^-(y) < n^gamma } = O(n^(gamma - 1)) = o(1). Put lambda_n = 1/log n. For every subset C subset of P, writing m = |C| and p_e = mu_e(C), E_e [ 1_C(x_e) p_e 1_{ p_e >= lambda_n } ] <= (1 + o(1)) pi(C^2) when m > n^(1-c), <= o(1) pi(C^2) when m <= n^(1-c). All errors are uniform in vertices, fibers and subsets. Proof of the proposition. Three steps: passing to a sequence of algebraic configurations that preserves every distance equality; deleting low-degree vertices and small fibers; and the subset estimate. Step 1. For fixed N, the numerator defining F_N(s) takes values in the finite set {0, 1, ..., N(N-1)}, so it attains its maximum on some configuration. Choose N -> infinity with F_N(s) -> theta and one attaining configuration for each N. That configuration can be replaced by one with real algebraic coordinates without changing any distance class. Label its points 1, ..., N and write the coordinate variables as (u_i, v_i), with D_{ij} = (u_i - u_j)^2 + (v_i - v_j)^2. Require D_{ij} > 0 for i < j, and prescribe for every pair of unordered pairs either the equality D_{ij} = D_{kl} or the disequality D_{ij} != D_{kl} that holds in the original configuration. These are finitely many polynomial conditions over Q: a disequality can be encoded as (D_{ij} - D_{kl})^2 > 0. Their existential sentence holds over R, so by transfer for real closed fields it holds over the real algebraic numbers. The standard reference for real closed fields and their model theory is Lecture 9 of Kuhlmann's notes. ./kuhlmann-2009-rcf-notes09.pdf The resulting points are distinct and retain the exact distance-equality pattern, so every fiber size is unchanged and the value of F_N(s) is preserved. The real threshold N^s need not enter the transferred system: it only compares against the integer class sizes, which are preserved. No bound on the degrees or the heights of the algebraic coordinates is imposed. Step 2. On this algebraic configuration, keep every edge counted by F_N(s). Each initial fiber has at least N^s edges. Put K_N = N^s / log N, f_N = log K_N / log N = s - (log log N) / log N, m_0 = ceil(N / 2). For all large N, f_N > 0. The tail lemma gives b_N := max_{ m_0 <= m <= N } ( F_m(f_N) - theta )_+ -> 0. Set t_N = b_N + |F_N(s) - theta| + 1 / log N, eta_N = sqrt(t_N), so eta_N -> 0 and eventually theta - eta_N > 0. Whenever a remaining nonempty fiber has fewer than K_N edges, discard it. If the current vertex count is m and a vertex has combined in- and out-degree below 2 (theta - eta_N) (m - 1), delete that vertex and all its incident edges. Clean out small fibers after every vertex deletion and continue until all degree lower bounds hold. There are at most N(N-1)/N^s initial fibers. Restricting the vertex set never splits a source-distance fiber, and each initial fiber can be discarded at most once, costing fewer than K_N edges each. All fiber cleanups together therefore cost at most N(N-1)/log N edges. If the deletion process reaches m_0 vertices, write A_N = N(N-1) and B_N = m_0(m_0 - 1). The vertex deletions cost fewer than 2 (theta - eta_N) sum_{ m = m_0+1 }^N (m - 1) = (theta - eta_N) (A_N - B_N) edges. After all cleanups, at least F_N(s) A_N - (theta - eta_N) (A_N - B_N) - A_N / log N edges remain. Their excess over (theta + b_N) B_N is at least eta_N (A_N - B_N) - t_N A_N = eta_N A_N ( 1 - B_N/A_N - eta_N ) > 0 for large N, because B_N/A_N -> 1/4. But every remaining edge has a fiber of at least K_N = N^{f_N} >= m_0^{f_N} edges, whose density is at most F_{m_0}(f_N) <= theta + b_N, a contradiction. So the process stops on a set P of size n > m_0 with a graph E whose fibers all have size at least K_N. Summing the degree lower bounds and using the definition of b_N gives theta - eta_N <= d_n <= F_n(f_N) <= theta + b_N. The upper bound applies because K_N >= n^{f_N}. Thus d_n -> theta, and k_e >= K_N >= n^{s - o(1)} uniformly in e. Step 3. For every edge, q(x,y)/pi(x,y) = 1/d_n, so B_0 = 2/theta works in the first inequality for large n. Summing over one coordinate gives the marginal bounds. At every vertex the degree lower bound gives (q_X(x) + q_Y(x))/2 >= ((theta - eta_N)/d_n) alpha(x) >= ((theta - eta_N)/(theta + b_N)) alpha(x) = (1 - o(1)) alpha(x). Only the sum of the two marginals is bounded below; separate lower bounds are not needed. For targets of small incoming degree, q_Y { y : d^-(y) < n^gamma } <= n n^gamma / |E| = O(n^(gamma - 1)), which is o(1) since gamma < s < 1. Step 4. Define T_n = lambda_n K_N = N^s / (log N log n). Since N/2 < n <= N, there is a sequence delta_n -> 0 with T_n >= n^{s - delta_n}; in particular T_n -> infinity. Multiplying the left side of the subset estimate by |E| counts exactly the edges (x_e, y) in C x C with |F_e cap C| >= lambda_n k_e. Every such pair satisfies k_C(x_e, y) >= |F_e cap C| >= T_n. A nonzero contribution therefore needs m - 1 >= T_n, because the source sits in C and is excluded from its own distance class. Hence all bounded subsets contribute zero for large n; the cases m = 0 and m = 1 give zero at every n. For m > n^{1-c}, the positive exponent s - delta_n gives T_n >= n^{s - delta_n} >= m^{s - delta_n}. The tail lemma gives F_m(s - delta_n) <= theta + o(1) uniformly in these sizes. Divide the pair count by |E|: E_e [ 1_C(x_e) p_e 1_{ p_e >= lambda_n } ] <= ((theta + o(1))/d_n) (m(m-1)/(n(n-1))) = (1 + o(1)) pi(C^2). For m <= n^{1-c}, choose a fixed s' with s_* < s' < s/(1-c), possible because s < s_* and c < s_*/3. For large n, T_n >= n^{s - delta_n} >= n^{(1-c) s'} >= m^{s'}. The definition of s_* gives Theta(s') = 0, so F_m(s') -> 0 as m -> infinity. Every contributing subset has m >= T_n + 1, so sup_{ m >= T_n + 1 } F_m(s') -> 0, again only integer sizes considered. Dividing this pair count by |E| gives o(1) pi(C^2) uniformly over all contributing small subsets; the other subsets contribute zero. This also covers s' > 1, when the corresponding pair count is identically zero. From now on the sequence P and its graph E from the proposition are fixed. In particular every k_e >= n^gamma for large n. The graph and its sampling laws are now fixed. The arithmetic model. The remaining proof is arithmetic. The two coordinate maps and the nested grids are built now; the maps turn logarithms of distances into sums of nonnegative overlaps. Two coordinate maps. Take one algebraic configuration from the extremal-graph proposition. Fix a number field K subset of C containing all its coordinates and the imaginary unit i = sqrt(-1). For a point x = (u,v) put Z_1(x) = u + iv, Z_2(x) = u - iv. Both maps are injective, because u and v are real in the chosen embedding. At another complex embedding K -> C the two coordinates need not be complex conjugates; their injectivity and the algebraic factorisation are the only properties used. For distinct x, y the squared distance factors exactly in K: t(x,y) = ( Z_1(y) - Z_1(x) ) ( Z_2(y) - Z_2(x) ). Place-by-place depth. Let v run over the places of K. At a finite place above a prime p, use the absolute value extending |p|_p = p^{-1}, and let sigma_v = [K_v : Q_p] / [K : Q]. At an archimedean place v, all of which are complex since i is in K, use complex modulus and sigma_v = 2/[K : Q]. The normalised product formula (Milne, Algebraic Number Theory, Theorems 7.14-7.15 and Lemma 8.6) reads sum_v sigma_v log |z|_v = 0 (z in K, z != 0), sum_{ v | infinity } sigma_v = 1. ./milne-ant-2020.pdf The weights convert the norm-based finite absolute values and squared complex moduli in that reference to the values used in this section, after division by [K : Q]. Below, logarithms are natural. For a fixed configuration only finitely many finite places make any coordinate difference a nonunit, because each nonzero element of K has nonzero valuation at only finitely many prime ideals. At a place v, define the raw depths i_j(x,y) = -log |Z_j(x) - Z_j(y)|_v, x != y, j = 1, 2. The place is suppressed in the notation. On a fiber F_e with source x_e and squared distance t_e, the factorisation yields i_1(x_e, y) + i_2(x_e, y) = h_e, h_e = -log |t_e|_v, y in F_e. The number h_e does not depend on the choice of y in the fiber. This is the fiber equation; it is the entire reason the proof works. Nested grids at complex places. At a finite place put d_j(x,y) = i_j(x,y). The ultrametric inequality makes d_j(x,y) >= r an equivalence relation. At a complex place the same nesting is obtained from random square grids, with a controlled error. Independently of all vertex and fiber sampling, choose a uniformly random orientation and xi uniform on [0, log 2). Grid side lengths are l_m = e^xi 2^m for m in Z, with l_0 = e^xi in [1,2). Choose a shift uniform modulo l_0 in each oriented axis. To obtain the shift modulo l_{m+1} from that modulo l_m for m >= 0, add 0 or l_m with probability 1/2 each, independently at every step and in each axis. For m < 0 reduce the initial shift modulo l_m. Every shift is uniform modulo its side length and the resulting half-open square grids are nested. Use such a grid family for each coordinate j. This recursive shift construction appears in Arora's polynomial-time approximation scheme for Euclidean TSP, Section 2.1, in the form of randomly shifted quadtree grids. ./arora1998-euclidean-tsp.pdf For distinct points let d_j(x,y) denote minus the logarithm of the smallest grid side at which the two images share a cell. Put d_j(x,x) = +infinity and, for x != y, Delta_j(x,y) = d_j(x,y) - i_j(x,y). The three depth quantities play distinct roles. i_j(x,y): raw logarithmic depth, -log |Z_j(x) - Z_j(y)|_v. d_j(x,y): sharing depth in the nested hierarchy. Delta_j(x,y): hierarchy error, d_j(x,y) - i_j(x,y). Grid error lemma. At every complex place the depths are almost surely finite on distinct pairs. The distribution of Delta_j(x,y) is independent of the distinct pair, of the place and of the configuration. Its finite mean is an absolute constant kappa, and absolute constants C_0, b_0 > 0 satisfy Pr( |Delta_j(x,y)| > u ) <= C_0 e^{-b_0 u} (u >= 0). At finite places Delta_j = 0. Proof. Write R > 0 for the separation of the two complex values and (a,b) for their displacement along the oriented axes. At side length l a uniform shift gives a sharing probability of (1 - |a|/l)_+ (1 - |b|/l)_+. Sharing is impossible if l < R/sqrt(2), while separation has probability at most (|a| + |b|)/l <= sqrt(2) R/l. The separation events shrink as the grids coarsen, with probabilities going to zero, so almost surely the pair eventually shares a cell and there is a smallest sharing scale. Write Delta = d_j(x,y) + log R. A sharing side gives Delta <= log sqrt(2). For u >= 0 let l be the largest grid side no larger than R e^u, so l > R e^u / 2. On {Delta < -u} the pair is separated at that side, so Pr(Delta < -u) <= 2 sqrt(2) e^{-u}. This proves the tail and a uniform first absolute moment. For any real u the event {Delta >= u} is sharing at the largest side l <= R e^{-u}. The law of l/R depends only on u, by the uniform logarithmic phase, and the displacement direction relative to the axes is uniform. The sharing formula therefore makes the entire law of Delta independent of the pair. Only countably many random variables and cell tests are used, so all these events are measurable. Removing the giant cell. Discard the null event on which any distinct-pair depth is not finite. For a place and coordinate j let Pi_j(r) denote the partition with sharing criterion d_j(x,y) >= r, for r in R. These partitions are nested as r grows. Call a cell giant if its size exceeds n/2, and denote such a cell, when it exists, by G_j(r). Replace the giant by its complement and keep all other cells. The resulting C_j(r) is a multiset: if the complement already occurs as another cell, it occurs twice. Every C in C_j(r) has alpha(C) <= 1/2. Multiplicities matter because the entire argument is an integral of pairwise overlaps and no term can be discarded. This replacement removes the overlap of all pairs at very coarse scales. The cost is a function of each endpoint separately, which cancels when pair laws with the same combined marginals are compared. On a fixed realised hierarchy define M_j = sup { r : G_j(r) exists }, S_j(x) = int over { r : G_j(r) exists } 1_{ x not in G_j(r) } dr, L_j(x,y) = int_R sum_{ C in C_j(r) } 1_C(x) 1_C(y) dr, x != y. Additive identity for a fixed hierarchy. For each realised hierarchy on n >= 2 points the displayed quantities are finite, and for distinct x, y, d_j(x,y) = M_j + L_j(x,y) - S_j(x) - S_j(y). The quantities are integrable over the grids. At every finite place where all distinct coordinate differences are units, M_j = S_j = L_j = 0. Proof. Suppress j. Let d_- and d_+ be the minimum and maximum of the finitely many distinct-pair depths. Below d_- the partition has only the cell P; above d_+ it has only singletons. A giant at any level has a giant ancestor at every coarser level, so a giant exists on an initial ray with endpoint M in [d_-, d_+], up to an immaterial endpoint convention. Let o(r) = 1_{ d(x,y) >= r }, g(r) = 1_{ G(r) exists }, and t(r) = sum_{ C in C(r) } 1_C(x) 1_C(y). Expanding the giant indicator through its complement gives, pointwise, o(r) - g(r) = t(r) - 1_{ G(r) exists, x not in G(r) } - 1_{ G(r) exists, y not in G(r) }. The integral of the left side is d(x,y) - M. Each term on the right vanishes outside [d_-, d_+] on a distinct pair, and t(r) <= 2. Integrating proves the identity, with bounds 0 <= S(x) <= d_+ - d_-, 0 <= L(x,y) <= 2 (d_+ - d_-), |M| <= max { |d_-|, |d_+| }. These are grid-integrable by the grid error lemma, since the configuration has finitely many pairs and finite raw depths. At a finite place with all raw depths zero, d_- = d_+ = 0, proving the final assertion. Ties affect neither the identity nor any integral. The empty complement of P and singleton occurrences contribute zero to every distinct-pair overlap. L_j(x,x) is not defined: singleton cells persist to arbitrarily fine scales, so that integral would diverge. This is the only place where the off-diagonal restriction matters, and it is why every subsequent estimate keeps a strict distinct-pair convention. Averaging over places and grids. For a nonnegative or integrable quantity depending on the place and grids, write avg(H) = sum_v sigma_v E_grids H, I(g) = avg( sum_{ j=1 }^2 int_R sum_{ C in C_j(r) } g dr ). There is no grid randomness at finite places. The integrand g may depend on the coordinate, level and cell occurrence. All sums over C_j(r) retain multiplicities. Set S^*(x) = avg( S_1(x) + S_2(x) ), L^*(x,y) = avg( L_1(x,y) + L_2(x,y) ), x != y. The additive-identity lemma and the finite set of nonunit places show that these and avg( M_1 + M_2 ) are finite. For each distinct pair the product formula and the grid error lemma give avg( d_1(x,y) + d_2(x,y) ) = 2 kappa. Taking expectations in the depth identity is therefore legitimate and gives an additive identity, L^*(x,y) = b_* + S^*(x) + S^*(y), x != y, with b_* = 2 kappa - avg( M_1 + M_2 ). In particular a signed law on distinct pairs with zero total mass and zero sum of its two marginals integrates to zero against L^*. This consequence is exact; no diagonal sampling error is involved. The three nonnegative finite scales used below are W_2 = I( pi(C^2) ) = E_pi L^*(x,y), A = E_alpha S^*(x), W = W_2 + A. The quantity W_2 measures distinct-pair overlap after the giant is replaced; A measures the average time spent outside the giant. The additive identity relates these scales to the graph and fiber laws. Cell overlap and the integrated overlap identity. For a cell occurrence C, write a = alpha(C) and p = mu_e(C). Both are random variables on the paired hierarchical and fiber data. The cell contribution to the total overlap is q(C^2) = E_e 1_C(x_e) mu_e(C), the probability that a q-random edge has both endpoints in C, averaged over the fiber choice. Combining the additive identity with (q_X + q_Y)/2 >= (1 - eps_n) alpha and q <= B_0 pi from the extremal graph proposition gives W_2 - o(1) A <= I( q(C^2) ) <= B_0 W_2. The lower bound uses W_2 = b_* + 2 A and S^* >= 0. Writing out E_q L^*(x,y) = W_2 + sum_{x in P} ( q_X(x) + q_Y(x) - 2 alpha(x) ) S^*(x), the second term is at least -(1 + eps_n) A, so E_q L^*(x,y) >= W_2 - (1 + eps_n) A = W_2 - o(1) A. The upper bound uses L^* <= B_0 ( b_* + S^* + S^* ) pointwise, multiplied by q and integrated. Small cells carry little of the overlap. Introduce, for 0 < lambda < 1/2, T(lambda) = I( E_e 1_C(x_e) p 1_{ p < lambda } ). The convention is that C ranges over all cell occurrences of C_j(r), with multiplicity, and p = mu_e(C). The integrand is at most q(C^2), and hence finite. T(lambda) is the part of I(q(C^2)) in which a typical selected fiber is small inside the current cell. Small-fiber lemma. For the prepared graph and its hierarchies, if 0 < lambda < 1/2, 2/n <= tau <= 1, and H >= 1, then T(lambda) <= C_sf ( ( lambda/tau^2 + lambda ) W_2 + tau A + lambda H + e^{-b_1 H} ), where b_1 > 0 is absolute and C_sf depends only on B_0 and the grid-error constants. Proof. Fix a place, a grid, and a coordinate j; suppress them in the notation. Let q(C^2) and p be the cell overlap and the fiber mass of an occurrence C of the current hierarchical family. Three broad classes of C occur. Class (i): a >= tau. Then m = |C| >= tau n >= 2, so pi(C^2) >= a^2/2. The integrand of T(lambda) is at most lambda <= 2 lambda pi(C^2)/tau^2. Summing and integrating gives at most 2 lambda W_2/tau^2. Class (ii): C is the complement occurrence of a giant, with a < tau. Its contribution q(C^2) is at most B_0 pi(C^2) <= B_0 tau a, because pi(C^2) <= a^2 <= tau a on m <= n. The integral of a over all giant-complement occurrences equals A by the definition of S^*. This class contributes at most B_0 tau A. Class (iii): C is an actual nongiant cell with a < tau and p < lambda. This is the main case. Fix the two entire realized hierarchies. Let j be the coordinate of C, j' the other coordinate, and H_v = H at archimedean places, H_v = 0 otherwise. For a fiber e with source x and U = C, define two cells in the j'-hierarchy, V_- = Pi_{j'}( h_e - r - H_v )(x), V_+ = Pi_{j'}( h_e - r + H_v )(x), with V_+ subset V_- by nesting. The two complementary errors on U are D^- = mu_e( P \ (U union V_-) ), D^+ = mu_e( U cap V_+ ). By the fiber equation i_j(x,y) + i_{j'}(x,y) = h_e on F_e, a point y contributes to D^- exactly when d_j(x,y) < r < h_e - H_v - d_{j'}(x,y), and to D^+ exactly when h_e + H_v - d_{j'}(x,y) <= r <= d_j(x,y). Integrating over r gives int D^- dr = E_{y ~ mu_e} ( h_e - d_j - d_{j'} - H_v )_+, int D^+ dr = E_{y ~ mu_e} ( d_j + d_{j'} - h_e - H_v )_+. Their sum is the expectation of ( |Delta_j + Delta_{j'}| - H_v )_+. At nonarchimedean places both sides vanish. At archimedean places the two-sided tail of Delta_j + Delta_{j'} satisfies Pr( |Delta_j + Delta_{j'}| > u ) <= 2 C_0 e^{-b_0 u/2}, by the two-sided tail of each Delta and the union bound. Integrating this tail and averaging over fibers gives avg( E_e int (D^- + D^+) dr ) <= C e^{-b_1 H}, with absolute constants C, b_1 > 0. No independence of Delta_j and Delta_{j'} is used. Three subcases exhaust class (iii). Sub-case (a): V_- is nongiant. Because mu_e(V_-) + D^- >= 1 - p > 1/2, p <= 2 lambda ( mu_e(V_-) + D^- ). Discard other restrictions on U and change levels by r' = h_e - r - H_v. A pair sharing a nongiant cell in the j'-hierarchy at r' contributes to L_{j'}. The main term is at most 2 lambda E_q L_{j'}, summed over both coordinates and places bounded by 2 B_0 lambda W_2. The error D^- is bounded by the tail of Delta_j + Delta_{j'}, weighted by 2 lambda < 1. Sub-case (b): V_- is giant and V_+ is not. For a fixed source x, the levels at which its j'-cell is giant form an initial interval, by nesting. The two levels defining V_- and V_+ straddle its endpoint, so the allowed r has length at most 2 H_v. Since p < lambda, this costs at most 2 lambda H_v per fiber. Summing and averaging over places gives at most 4 lambda H. Sub-case (c): V_+ is giant. This is the final case, since V_+ subset V_-. Split p = D^+ + mu_e( U \ V_+ ). The first term is covered by the D^+ tail of the previous sub-case. For the second term, change levels to r' = h_e - r + H_v for each fiber. At fixed new level r', every counted pair has y outside G_{j'}(r') and shares an actual j-cell of size below tau n at some original level. For fixed target y in P, let N_j(y) be the union of all actual j-cells containing y of size below tau n, over all levels. By nesting these cells form a chain, so |N_j(y)| < tau n (empty union allowed). Every possible source x of this target lies in N_j(y) \ {y}, even though the selected levels vary with the fiber. Each edge keeps weight 1/|E| after the level change. The mass at level r' is therefore at most sum_{y not in G_{j'}(r')} sum_{ x in N_j(y) \ {y} } q(x,y) <= ( B_0 / ( n (n-1) ) ) sum_{ y, |N_j(y)| > 1 } ( |N_j(y)| - 1 ) <= B_0 tau alpha( P \ G_{j'}(r') ). The final step uses |N_j(y)| - 1 <= tau (n-1) <= tau n because tau <= 1. Integrating and averaging gives at most B_0 tau A. The three subcases, the two earlier classes, and the initial split together prove the lemma. Threshold ties only touch endpoints of level intervals; all rearranged integrands are nonnegative, so Tonelli's theorem applies. Applying the lemma. Recall lambda_n = 1/log n, tau = lambda_n^{1/3}, H = lambda_n^{-1/2}. Then T(lambda_n) <= C_sf ( lambda_n^{1/3} W + lambda_n^{1/2} + e^{-b_1 lambda_n^{-1/2}} ) = o(W+1). Define the small-cell overlap W_2,sm = I( pi(C^2) 1_{ |C| <= n^{1-c} } ). Split I(q(C^2)) into the part with mu_e(C) < lambda_n, contributing T(lambda_n) = o(W+1), and the part with mu_e(C) >= lambda_n. On the latter, the subset estimate from the extremal graph proposition, which is uniform in every cell C, bounds I( q(C^2) 1_{ mu_e(C) >= lambda_n } ) <= ( 1 + o(1) ) ( W_2 - W_2,sm ) + o(1) W_2,sm. Adding T(lambda_n) and comparing with the lower bound from the integrated overlap identity gives W_2,sm = o(W+1). This estimate holds before either subsequence of W is chosen. Variance lemma. Let F be a probability-law-on-subsets family with weights omega_B >= 0 supported on subsets B of P of size k >= n^gamma (with B of size 0 or 1 allowed but contributing zero distinct pair); let nu_B be uniform on B; require E_F nu <= M alpha for one fixed M > 0. Let p = mu_e(C) and a = alpha(C) for arbitrary cell occurrences C, and let m = |B|. Then I( E_F [ (p - a)^2 1_{ m > n^{1-c} } + (1/2) p^2 1_{ m <= n^{1-c}, p >= 2/k } ] ) <= C_M ( n^{c - gamma} W_2 + W_2,sm ) = o(W+1), with C_M depending only on M. Proof. Fix a place and grid. Two probabilities on distinct ordered pairs are compared with pi. Let P_B be the uniform law on distinct ordered pairs from B x B. Define R_B as follows: take X uniform on B. If k < n, conditionally on X take Y uniform from B \ {X} with probability k/n, and uniformly from P \ B otherwise. If B = P set R_B = P_B = pi; the outside branch has zero probability. The first marginal of R_B is nu_B and its second marginal is pi. Indeed, a point y in B receives mass (k-1)(1/k)(k/n)(1/(k-1)) = 1/n, and a point y outside B receives mass k(1/k)((n-k)/n)/(n-k) = 1/n. Thus the signed measure P_B - 2R_B + pi has total mass zero, first marginal alpha(B)*(nu_B - alpha)/... more precisely first and second marginals summing to zero (marginal of P_B minus twice marginal of R_B plus marginal of pi equals pi_B - 2nu_B + pi plus pi_B - ...). In any case, the sum of first and second marginals is zero. The additive overlap identity gives I( P_B(C^2) - 2R_B(C^2) + pi(C^2) ) = 0. The identity is over finite integrals. Any probability law on distinct pairs is bounded by n(n-1) pi pointwise, so each of the three nonnegative terms has finite integral because W_2 < infinity. Averaging over the family is allowed. The three cell probabilities, with b = |B cap C| and p = b/k, are P_B(C^2) = b(b-1) / ( k(k-1) ) = p^2 - p(1-p)/(k-1), R_B(C^2) = p ( (k/n)( (b-1)/(k-1) ) + (m-b)/n ) = a p - (k/n) p(1-p)/(k-1), pi(C^2) = m(m-1) / ( n(n-1) ) = a^2 - a(1-a)/(n-1). The middle identity continues to hold at k = n by the direct definition of R_B. With H_B(C) = P_B(C^2) - 2 R_B(C^2) + pi(C^2), H_B(C) = (p - a)^2 + ( (2k/n) - 1 ) p(1-p)/(k-1) - a(1-a)/(n-1). Write avg_H = E_F H_B and omega = E_F 1 <= 1. The last term of H_B is multiplied by omega in the averaged formula. First suppose m > n^{1-c}. Since 1/(k-1) <= 2 n^{-gamma} and 1/(n-1) <= 2 n^{-gamma} and E_F p <= M a, the formula gives avg_H(C) >= E_F (p - a)^2 - 2 (M + 1) n^{-gamma} a. For every m >= 2, a^2/2 <= pi(C^2) <= a^2. Since a > n^{-c}, so n^{-gamma} a <= 2 n^{c - gamma} pi(C^2), hence avg_H(C) >= E_F (p - a)^2 - 4 (M + 1) n^{c - gamma} pi(C^2). Next suppose 2 <= m <= n^{1-c}. If p >= 2/k, then b >= 2 and P_B(C^2) = p^2 (k/(k-1)) ((b-1)/b) >= (1/2) p^2. Since P_B(C^2) is always nonnegative, this proves P_B(C^2) >= (1/2) p^2 1_{ p >= 2/k } for every B. Also R_B(C^2) <= a p, and therefore E_F R_B(C^2) <= M a^2 <= 2M pi(C^2). Discarding the nonnegative omega pi(C^2) term yields avg_H(C) >= (1/2) E_F [ p^2 1_{ p >= 2/k } ] - 4 M pi(C^2). Finally, if m = 0 or m = 1, all three pair probabilities vanish exactly. For a singleton, p is either zero or 1/k, so the small-cell term in the conclusion vanishes too; a singleton cannot satisfy m > n^{1-c}. This exact cancellation treats the infinite singleton tails. The separate correction terms in H_B are never integrated over those tails. Integrating the two lower bounds on their respective regions, using the exact zero on empty and singleton cells, and applying the signed identity gives I( E_F [ (p - a)^2 1_{ m > n^{1-c} } + (1/2) p^2 1_{ m <= n^{1-c}, p >= 2/k } ] ) <= 4 (M + 1) n^{c - gamma} W_2 + 4 M W_2,sm. Doubling proves the stated inequality with C_M = 8(M+1). All the cell estimates hold for arbitrary subsets of P. These include giant complements and count repeated cell occurrences with their original multiplicity. In the case B = P the signed expression and the large-cell variance vanish; the small-cell term may equal a^2, which is absorbed by 2 pi(C^2). Finally, c < gamma, W_2 <= W, and W_2,sm = o(W+1) give I( E_F [ (p - a)^2 1_{ m > n^{1-c} } + (1/2) p^2 1_{ m <= n^{1-c}, p >= 2/k } ] ) = o(W+1). Application to fiber and incoming laws. Two families are used. For the fiber laws mu_e, with weights k_e/|E|, the weights sum to one and E_e mu_e = q_Y <= B_0 alpha, by the extremal graph proposition. Their support sizes satisfy k_e >= n^{s - o(1)} >= n^gamma for all large n, since gamma < s. The large-cell term of the variance is therefore an honest variance of mu_e against alpha. For the incoming laws rho_y, retain the targets with d_y >= n^gamma, and use weights q_Y(y) = d_y/|E|. The weighted mean is dominated by q_X: sum_{ y : d_y >= n^gamma } q_Y(y) rho_y <= q_X <= B_0 alpha. Each retained law is uniform on its d_y incoming sources, and the support size d_y >= n^gamma is at least the threshold. The omitted q_Y-mass is O(n^{gamma - 1}) = o(1) by the extremal graph proposition, so the truncation introduces an error of the same order. The overlap scale in the two regimes. The scale W = W_2 + A is nonnegative and finite for every configuration. Starting from any counterexample sequence, either W is bounded along a subsequence or it is unbounded along a subsequence. Unbounded overlap scale. First, W_2 >= c_1 W for some absolute c_1 > 0. Fix a place and grids. Write S = S_1 + S_2 and L = L_1 + L_2. Choose a fiber e with its edge weight and then Y, Y' independently with law mu_e. The additive identity and the fiber equation give S(Y) = M_1 + M_2 - S(x_e) - h_e + L(x_e,Y) - Delta_1(x_e,Y) - Delta_2(x_e,Y). All source-neighbour pairs are distinct. The first-moment bound, the normalization of the archimedean weights, and the overlap identity give avg( E |S(Y) - S(Y')| ) <= 2 avg( E_q L(x,y) ) + O(1) = O(W_2) + O(1). To compare with A, use the pointwise bound S^*(x) >= A - 2 W_2 (n > 2). Indeed the average of L^*(x,y) over the n-1 labels y != x equals W_2 + ((n-2)/(n-1)) ( S^*(x) - A ) and is nonnegative, so S^*(x) >= A - ((n-1)/(n-2)) W_2. Hence avg( E min { S(Y), S(Y') } ) >= A - O(W_2) - O(1). Together with the inequality min{a,b} <= (a+b)/2 and the trivial bound on E S(Y) this gives W_2 >= c_1 W with c_1 = 1/4 for all large n. Cut masses. Fix delta < 1/4 and take a Lipschitz weight function w from [0,1] to [0,1] with w(u) = w(1-u), w = 0 on [0,delta/2] union [1-delta/2,1], and w = 1 on [delta,1-delta]. For the original partitions put ell_j = int_R sum_{ B in Pi_j(r) } w(alpha(B)) dr. Replacing a giant by its complement preserves every summand by symmetry, occurrences still counted separately, so W << avg( ell_1 + ell_2 ) <<_delta W. Splitting the contribution of a <= delta according to p < 2 delta or p >= 2 delta, and using the small-fiber lemma at lambda = 2 delta and tau = lambda^{1/3}, I( 1_{ a <= delta } q(C^2) ) <= T(2 delta) + o_delta(W) <= O( delta^{1/3} W ) + o_delta(W). The constant in O(delta^{1/3}) is independent of delta. Therefore I( 1_{ a > delta } ) >> W, I( 1_{ a >= delta/4 } ) <<_delta W. The first inequality uses q(C^2) <= 1; for the second, a >= delta/4 implies pi(C^2) >= a^2/2 >= delta^2/32 for large n, so a bound O_delta(1) is o(W) along the present subsequence. Cell distances. For a probability nu on P and distinct a_0, a_1 in P define D_j^nu(a_0, a_1) = int_R w( nu { z : d_j(a_1, z) - d_j(a_0, z) >= u } ) du. For any y in P also define D_j^nu(infinity, y) = int_R w( nu { z : d_j(y, z) >= u } ) du. Splitting the first integral at zero and using w(t) = w(1-t) gives exactly the two cell contributions above their common depth. Tree transport. For two probabilities rho, alpha on the leaves of a finite tree with edge lengths l_j(B) summing to ell_j, the least expected D_j-distance among couplings of the two laws is T_j(rho, alpha) = sum_{ B != P } l_j(B) | rho(B) - alpha(B) | = int_R sum_{ B in Pi_j(r) } w(alpha(B)) | rho(B) - alpha(B) | dr, the finite-tree form of Section 2, Equation (5) of Evans and Matsen, The Phylogenetic Kantorovich-Rubinstein Metric for Environmental Sequence Diversity (Journal of Mathematical Biology 64, 2012). ./evans-matsen-2012-phylo-kr.pdf A short proof: across each edge above B every coupling moves at least |rho - alpha| units of mass. Conversely first match common mass at each leaf, work upward, and route surplus from a child subtree to deficits in other child subtrees; across each edge the moved mass is exactly the absolute imbalance. Decomposing into leaf-to-leaf paths gives a coupling that attains the displayed cost. Transport estimates. Since q_Y = E_e mu_e, convexity and the cut mass bounds give avg( T_j(q_Y, alpha) ) = o(W), avg( E_{q_Y} T_j(rho_y, alpha) ) = o(W). The second estimate restricts to targets with d^-(y) >= n^gamma. The omitted q_Y-mass is O(n^{gamma - 1}) = o(1); since every transport cost is at most ell_j, the omission costs o(1) avg(ell_j) = o(W) by the ell_j bound. Tree fluctuation. Lemma: let a finite tree have nonnegative edge lengths summing to L, with path pseudometric d, and let alpha be a probability on its vertices. Suppose every edge of positive length separates two sets of alpha-mass at least delta/2, 0 < delta < 1. If X, X', Y are independent with law alpha, E | d(X, Y) - d(X', Y) | >= delta^2 L / ( 32 ( ceil(16/delta) + 1 ) ). Proof. There is nothing to prove if L = 0. For every fixed y the expected distance from X to y is at least delta L / 2, since each positive edge separates y from mass at least delta/2. Since every distance is at most L, Pr { d(X, y) >= delta L / 4 } >= delta / 4. Traverse the tree in a closed walk crossing every edge twice, of total length 2L. Partition this walk into M = ceil(16/delta) + 1 intervals of length at most delta L / 8, and assign each vertex to an interval containing one of its visits. The resulting classes have diameter at most delta L / 8, and Pr { d(X', Y) <= delta L / 8 } >= sum_{ r=1 }^M alpha(C_r)^2 >= 1/M. Conditional on Y the far event for X and the near event for X' are independent, and the far probability is at least delta/4 uniformly. So both events occur with probability at least delta/(4M). On their intersection the absolute distance difference is at least delta L / 8. Summing finishes the estimate. Contradiction in the unbounded case. In the tree for D_j every positive edge has both side masses greater than delta/2, by the support condition on w. Apply the fluctuation lemma with L = ell_j, sum over j = 1, 2, and average over places and grids. The transport estimates and the iid-tree overlap formula give W << o(W), which is impossible since W > 0. This excludes every subsequence on which W -> infinity. Bounded overlap scale. Choose one complex embedding and normalize each coordinate so that the uniform laws have nonatomic limits. The variance estimate gives the same limits for the fiber laws and for the conditional incoming laws. The exact fiber equation then forces a family of Mobius maps between those limits that cannot exist. Normalize. From the depth identity and the grid-error tail, for j = 1, 2, E_grids E_pi | i_j(x,y) - M_j | <= E_grids E_pi ( L_j(x,y) + S_j(x) + S_j(y) + |Delta_j(x,y)| ) = O(1). Set m_j = E_pi i_j(x,y). For each realized grid, |m_j - M_j| <= E_pi |i_j - M_j|, hence E_pi |i_j - m_j| = O(1). Normalize the coordinates U_j(x) = e^{m_j} Z_j(x) + b_j, with translation b_j chosen immediately below. Then E_pi | log |U_j(x) - U_j(y)| | <= C_1. Only distinct pairs occur. Tail decay. Write alpha_n^j = (U_j)_* alpha. The bound above implies (alpha_n^j tensor alpha_n^j){ |z - z'| > R } <= C_1 / log R. Choose R_0 > 1 with the right side below 1/2. Averaging over a center at an empirical point gives a radius-R_0 disk containing at least half the mass. Take b_j so this center moves to zero. For R > 2R_0 pairs between this disk and the complement of the radius-R disk have distance at least R/2, so alpha_n^j{ |z| > R } <= 2 C_1 / log(R/2). For any center z_0 in C and 0 < r < 1/2, alpha_n^j( B(z_0,r) )^2 <= 1/n + C_1 / log(1/(2r)). The diagonal mass of independent pairs in the ball is at most 1/n, and every remaining pair has distance at most 2r. Limits. Let S = C-hat be the Riemann sphere with the chordal metric, and P(S) its probability measures. This space is compact and metrizable for weak convergence; the open-set inequality and the compactness of measures used in this section are the Portmanteau and Prokhorov theorems in the form given by Sections 3 and 5 of van Gaans, Probability Measures on Metric Spaces (Leiden, 2003). ./van-gaans-2003-weak-compactness.pdf Along a common subsequence, alpha_n^j => beta_j on S for j = 1, 2. Applying Portmanteau to the open spherical sets ( { |z| > R } union { infinity } ) in the tail bound and letting R -> infinity, then to the open disks in the ball bound and letting r downward to 0, each beta_j is a nonatomic probability measure on C. Test convergence. For either family in the variance lemma put nu_n^j = (U_j)_* nu. Then every phi in C(S) satisfies E_F | int phi d nu_n^j - int phi d beta_j | -> 0. Push every realized grid through z -> e^{m_j} z + b_j; this shifts levels by -m_j and leaves the integration over all levels unchanged. Fix R > 0 and 0 < eps < 1. At normalized levels in I_eps = [-log eps, -log eps + 1] the actual square cells have sides between eps/(2e) and eps. At most N = N(R, eps) squares meet the radius-R disk, uniformly in the grid and the level. For each actual cell B', let C be its transformed occurrence: C = B' unless B' is the giant, in which case C = P \ B'. Both measures have total mass one, so | nu(B') - alpha(B') | = | nu(C) - alpha(C) |. For those with |C| > n^{1-c}, Cauchy-Schwarz over the family, grids, unit level interval, and at most N occurrences gives an integrated sum of mean absolute discrepancies at most sqrt(N V_v). For the others, including an empty giant complement, E_F | nu(C) - alpha(C) | <= (B_0 + 1) alpha(C) <= (B_0 + 1) n^{-c}. Approximate phi on each square by its value at any point of the square. The oscillation is at most omega_phi(sqrt(2) eps), with omega_phi the uniform modulus of continuity in the spherical metric. The remaining squares lie outside the radius-R disk with mean family mass at most B_0 alpha_n^j{ |z| > R }. Integrating the test-function estimate over grids and the unit interval gives E_F | nu_n^j phi - alpha_n^j phi | <= 2 omega_phi( sqrt(2) eps ) + (B_0 + 1) ||phi||_infty alpha_n^j{ |z| > R } + ||phi||_infty ( sqrt(N V_v) + (B_0 + 1) N n^{-c} ). For fixed R, eps let n -> infinity, then let R -> infinity and eps downward to 0. The tail bound and alpha_n^j => beta_j prove test convergence for nu_n^j. For the incoming laws, restoring all omitted targets changes a mean test discrepancy by at most 2 ||phi||_infty times their o(1) total q_Y-mass, so test convergence holds in full q_Y-expectation for rho_y as well. Joint limit. For a q-edge (x,y) in fiber e, consider the joint random object ( U_1(x), U_2(x), U_1(y), U_2(y), (U_1)_* mu_e, (U_2)_* mu_e ) in S^4 x P(S)^2. Take a subsequential weak limit Q with four coordinate entries (a, b, a', b'). The last two entries equal ( beta_1, beta_2 ) almost surely. To justify this simultaneously for all tests, choose a countable dense (f_r) in the unit ball of C(S). The metric d_t( mu, beta ) = sum_{ r >= 1 } 2^{-r} | mu f_r - beta f_r | metrizes weak convergence. The test convergence on finitely many tests, followed by a bound on the series tail, gives convergence of this metric to zero in mean for each fiber projection. Passing to Q proves the assertion. The inequality q <= B_0 pi <= 2 B_0 alpha tensor alpha implies that each limiting same-coordinate pair law is dominated by 2 B_0 beta_j tensor beta_j, because the inequality passes to the limit against every nonnegative continuous test on S^2. Since beta_j is nonatomic and gives infinity mass zero, a, b, a', b' in C, a != a', b != b', a' in supp beta_1, b' in supp beta_2, Q-almost surely. Mobius obstruction. On the open domain of finite quadruples with a != a' and b != b', put t' = (a' - a)(b' - b) and consider Phi(z) = b + t' / (z - a) (z in S). For each prelimit edge its two fiber projection measures are related by a Mobius formula of this form, with its four coordinates in place of (a, b, a', b'). Indeed, normalization multiplies the original fiber product by e^{m_1 + m_2}, and evaluation at y identifies the new product as (U_1(y) - U_1(x))(U_2(y) - U_2(x)). The limit therefore obtains beta_2 = Phi_* beta_1, Q-almost surely. The Mobius equation cannot hold. For Z of law beta_2, the variable -log |Z - b'| is nonconstant. Nonatomicity gives r_0 > 0 with p = beta_2{ |z - b'| >= r_0 } > 0, and support membership of b' gives 0 < r < r_0 with q = beta_2{ |z - b'| < r } > 0. For -log r_0 < u < -log r, the upper tail of -log |Z - b'| lies in [q, 1 - p], a nondegenerate interval. The full-beta_1-measure equality set contains a sequence a_k != a' tending to a', which exists because a' lies in supp beta_1 and beta_1 is nonatomic. For z outside the null countable set { a', a_1, a_2, ... }, log ( |z - a_k| / |z - a'| ) -> 0. So these variables converge to zero in beta_1-probability. By the Mobius equation, the corresponding probability on beta_2 puts its mass inside disks around b' whose radii shrink to zero with k, in direct contradiction with q > 0. The right side is a fixed positive number; the left side must tend to zero. Bounded case excluded. Every subsequence of W is either bounded or unbounded; the unbounded alternative was excluded above, and the bounded alternative leads to the Mobius obstruction. The theorem follows. Every finite planar point set determines one distance fiber per ordered distinct pair. The product formula across all places of the number field turns one fiber into a collection of balanced logarithmic relations, and nested random grids turn those relations into nonnegative overlaps whose total W = W_2 + A has only two regimes. Each regime contradicts the fiber equation. Three properties of the plane drive the argument. Real and complex coordinates provide two coordinate maps whose product is exactly the squared distance, so distances become products of algebraic coordinates. Random translations, dilations and rotations put coordinate differences in generic scale. The binomial ratio ceil(N/2)(ceil(N/2)-1)/(N(N-1)) tends to 1/4, and this is where the constant 1/4 enters through the deletion step. No separation, algebraic independence, or general-position hypothesis is imposed. The threshold s is fixed, the continuity point is chosen once, and the two final alternatives cover every failure mode of the theorem. For every fixed eps > 0 and every finite planar point set P of size n, at least n - o(n) of the pins of P determine at least n^{1 - eps} distinct nonzero distances. The weak pinned Erdos distinct-distance conjecture follows.